math education
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enCelebrating Role Models in Science & Engineering Achievement: Sonya Kovalevsky
https://scienceblogs.com/usasciencefestival/2014/01/22/celebrating-role-models-in-science-engineering-achievement-sonya-kovalevsky
<span>Celebrating Role Models in Science & Engineering Achievement: Sonya Kovalevsky</span>
<div class="field field--name-body field--type-text-with-summary field--label-hidden field--item"><p><strong>Sonya Kovalevsky – Russian-born Mathematician</strong></p>
<p><em>One of the world's best mathematicians of her era; established first major result in general theory of partial differential equations; first modern European woman appointed to full professorship; advocate of women's rights</em></p>
<p style="text-align: center;"><a href="https://scienceblogs.com/files/usasciencefestival/files/2014/01/Sonya-Kovalevsky-2.jpg"><img class="aligncenter wp-image-2104" alt="Sonya Kovalevsky 2" src="https://scienceblogs.com/files/usasciencefestival/files/2014/01/Sonya-Kovalevsky-2.jpg" width="440" height="421" /></a></p>
<p>Sonya Kovalevsky (also known as Sofia Kowalevski) was born in Russia in 1850 and became a noted mathematician in spite of a father who "had a horror of learned women," according to historical accounts. As a young woman, she could study math and physics only in secret. She married a man she did not love just to get away from her father and obtain a formal education.</p>
<p>She grew up a member of Russia's privileged social class. Her father was a military officer and a land holder; her mother, the granddaughter of a famous Russian astronomer, was an accomplished musician. The family lived comfortably on a country estate, where Sonya, her sister and brother were brought up by a nanny until their education was taken over by governesses and private tutors.</p>
<p>By the age of thirteen Sonya showed an unusual ability and enthusiasm for algebra and geometry. But her father believed that there was no need nor place for learned women, so he put a stop to further mathematical instruction. Secretly, Sonya borrowed an algebra book from one of her tutors and continued to study "under the covers" at night. About a year later a neighbor, who was a professor of science at a nearby school, gave the family a copy of an elementary physics book he'd written. When Sonya tried to read the section on optics, she bumped into trigonometry, a subject she had never heard of. To make sense of some of the derivations, she substituted "a chord for the mysterious sine," and everything worked for small angles.</p>
<p>The neighbor-professor was so impressed that Sonya had independently rediscovered the method by which the concept of sine had developed historically, that he tried to persuade Sonya's father to arrange serious training in mathematics for her. It took her father four years to agree to let her take private lessons in analytic geometry and calculus in St. Petersburg. She mastered these subjects quickly over one winter, much to the astonishment of her professor.</p>
<p><strong>Why She's Important:</strong> She was the first major Russian female mathematician, responsible for important original contributions to analysis, differential equations and mechanics, and the first woman appointed to a full professorship in Northern Europe. She was also one of the first women to work for a scientific journal as an editor.</p>
<p>Sonya was not only a great mathematician, but also a writer and advocate of women's rights in the 19th century. It was her struggle to obtain the best education available which began to open doors at universities to women. In addition, her ground-breaking work in mathematics made her male counterparts reconsider their archaic notions of women's inferiority to men in such scientific arenas.</p>
<p>But before conquering these frontiers, she had to overcome personal challenges in her own family. For example, as a young woman, she wanted desperately to pursue a career as a doctor or chemist ("to be of use"), her family would not allow their single daughters to go abroad. So in 1868 Sonya and her older sister arranged "fictitious marriages", or marriages of convenience to radical compatriots. Sonya married a promising young paleontologist, Vladimir Kovalevsky, who would contribute to the substantiation of Darwin's new and controversial theory of evolution, and (although not in love with him) moved to Heidelberg, Germany to study science and mathematics.</p>
<p>After two years of mathematical studies at Heidelberg under such teachers as Hermann von Helmholtz, Gustav Kirchhoff and Robert Bunsen, she moved to Berlin, where she had to take private lessons from famous mathematician Karl Weierstrass, as the university would not even allow her to audit classes. In 1874 she presented three papers—on partial differential equations, on the dynamics of Saturn's rings and on elliptic integrals —to the University of Göttingen as her doctoral dissertation. With the support of Weierstrass, this earned her a doctorate in mathematics summa cum laude, bypassing the usual required lectures and examinations.</p>
<p>She thereby became the first woman in Europe to hold that degree. Her paper on partial differential equations contains what is now commonly known as the Cauchy-Kovalevski theorem, which gives conditions for the existence of solutions to a certain class of those equations.</p>
<p><strong>Other Achievements:</strong> After the suicide death of her husband, Sonya was offered a teaching position at Stockholm University. After only a year, she was appointed to full professor (the first woman in Northern Europe to be named to such a post) and published her research on light refraction. In 1885 Kovalevsky was appointed chair of mechanics. Throughout her life Kovalevsky was also very interested in literature and pursued writing novels, a play, and radical political books such as A Nihilist Girl, and her memoirs.</p>
<p>But mathematics remained her main passion. In 1888 she won the prestigious Prix Bordin competition sponsored by the French Academy of Science. Her brilliant career was cut short, however, when she died on February 10, 1891, of pneumonia at age 41 in Sweden.</p>
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<span><a title="View user profile." href="https://scienceblogs.com/author/carlyo" lang="" about="https://scienceblogs.com/author/carlyo" typeof="schema:Person" property="schema:name" datatype="" xml:lang="">carlyo</a></span>
<span>Tue, 01/21/2014 - 18:55</span>
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Tue, 21 Jan 2014 23:55:44 +0000carlyo70562 at https://scienceblogs.comMath literacy is so important, but don't take my word for it.
https://scienceblogs.com/sciencewoman/2009/10/29/math-literacy-is-so-important
<span>Math literacy is so important, but don't take my word for it.</span>
<div class="field field--name-body field--type-text-with-summary field--label-hidden field--item"><p><img src="http://scienceblogs.com/sciencewoman/wp-content/blogs.dir/256/files/2012/04/i-9dc84d4d9156dccb30d5f62466b4219a-swblocks.jpg" alt="i-9dc84d4d9156dccb30d5f62466b4219a-swblocks.jpg" />There's a few days left in our October <a href="http://www.donorschoose.org/donors/viewChallenge.html?page=1&max=4&id=24200&category=111">DonorsChoose challenge</a>, and even after that there are many more great projects out there waiting for our help. </p>
<p>A few weeks ago, wonderful educator-science-historian-cultural-studies-expert-mother-blogger <a href="http://lesliemadsenbrooks.com/">Leslie Madden-Brooks</a> responded to a plea to help fund some projects, and I was deeply moved by what she wrote to the classroom, so I wanted to share it with you...</p>
<blockquote><p>I gave to this project because I had such a tough time learning math, and I wish I had been able to develop this kind of mathematical and critical thinking through reading interesting authors. I enjoy science tremendously, but I had to stop taking these classes early in college because I couldn't do the math required in them. I don't want any students to have that same lifelong handicap.</p></blockquote>
<p>Wow. I have so much respect for Leslie for admitting that and for trying to help some current elementary kids avoid the same dilemma. If Leslie's comment struck a nerve with you, consider helping one of these projects: <a href="http://www.donorschoose.org/donors/proposal.html?id=303632&challengeid=24200"> Math Literature Books Needed</a> for 3-5th graders in Michigan (needs just $99); or <a href="http://www.donorschoose.org/donors/proposal.html?id=328818&challengeid=24200">Math Read Alouds</a> for South Carolina 3-5th graders with learning disabilities (needs just $74).</p>
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<span><a title="View user profile." href="https://scienceblogs.com/author/sciencewoman" lang="" about="https://scienceblogs.com/author/sciencewoman" typeof="schema:Person" property="schema:name" datatype="" xml:lang="">sciencewoman</a></span>
<span>Thu, 10/29/2009 - 01:43</span>
Thu, 29 Oct 2009 05:43:43 +0000sciencewoman130937 at https://scienceblogs.comTeaching Multiplication: Is it repeated addition?
https://scienceblogs.com/goodmath/2008/07/25/teaching-multiplication-is-it
<span>Teaching Multiplication: Is it repeated addition?</span>
<div class="field field--name-body field--type-text-with-summary field--label-hidden field--item"><p> I've been getting peppered with requests to comment on a recent argument that's<br />
been going on about math education, particularly with respect to multiplication.<br />
We've got <a href="http://www.maa.org/devlin/devlin_06_08.html">a fairly prominent guy named Keith Devlin ranting that<br />
"multiplication is not repeated addition"</a>. I've been getting mail from both<br />
sides of this - from people who basically say "This guy's an idiot - of<br /><em>course</em> it's repeated addition", and from people who say "Look how stupid<br />
these people are that they don't understand that multiplication isn't repeated<br />
addition".</p>
<p> In general, I'm mostly inclined to agree with him, with some major caveats. But since he sidesteps the real fundamental issue here, I'm rather annoyed with him.</p>
<!--more--><p> You see, the argument isn't really about multiplication, but about math education. The argument isn't really about whether multiplication is repeated addition - it's about whether or not we should <em>teach kids</em> to understand multiplication as repeated addition. And that's a tricky question, because the answer is both yes <em>and</em> noe.</p>
<p> Is multiplication repeated addition? Sometimes, it is. But multiplication isn't <em>just</em> repeated addition. It includes cases where it makes sense to talk about it as repeated addition, and also cases where it doesn't.</p>
<p> What's exponentiation? Is it repeated multiplication? Sometimes. And sometimes it isn't. Try to give me a simple definition of exponentiation, which is understandable by a fifth or sixth grader, which <em>doesn't</em> at least start<br />
by talking about repeated multiplication. Find me a beginners textbook or<br />
teachers class plans that explains exponentiation to kids without at least starting with something like "5<sup>2</sup>=5×5, 5<sup>3</sup>=5×5×5."</p>
<p> With respect to multiplication, it's the same question, only with even younger kids: how do you explain multiplication to a third grader? How can you start to tell a kid about 2×2=4 and 2×3=6 without showing them that 2×2 = 2+2, and 2×3 = 2+2+2?</p>
<p> Multiplication isn't really a simple thing. What mathematicians mean by multiplication is, roughly, one of the two fundamental operations over the field of real numbers. Outside of the realm of abstract math, multiplication actually has<br />
multiple meanings, which each work in different contexts. But they're all concrete<br />
applications derived from the fact that multiplication is the second field operation<br />
in the field of real numbers.</p>
<p> But how are you going to explain that two a third grader?</p>
<p> Just think of one of the classic word problems that every kid sees in second or third grade when they start doing multiplication. Every kid in class has three apples; so how many apples does the class have?</p>
<p> When you're using that problem, repeated addition makes excellent sense. It also<br />
matches the mechanics of what the kids are doing. So it's a good intuitive way to<br />
get them started on understanding multiplication. It's not the whole picture - but it's an initial intuition that provides some concrete handle to grab on to.</p>
<p><img src="http://scienceblogs.com/goodmath/wp-content/blogs.dir/476/files/2012/04/i-36362faff4af5bc72ec91ebe8614149f-triangle.png" alt="i-36362faff4af5bc72ec91ebe8614149f-triangle.png" /></p>
<p> Of course, pretty soon, you have to break that intuition at least a little bit - because there are plenty of places where repeated addition just doesn't really make sense. Look at the figure over to the side. There's a triangle with a base five<br />
inches long, and it's two inches high, with the highest point being three inches in. What's the area of that triangle? 1/2 base×height, in square inches. How can you describe that by repeated addition? </p>
<p> For the triangle, you can do a geometric explanation of multiplication. The two numbers being multiplied are the sides of a rectangle, and multiplying is creating the area inside the rectangle. You can use that intuition to explain the area of a triangle, by showing how to create a rectangle by cutting the triangle into pieces, and re-arranging them. That gives you a geometric intuition about multiplication.</p>
<p> But neither of those is particularly good for explaining how multiplication can tell you what 3/5ths of $25 is.</p>
<p> So the real question isn't "Is multiplication repeated addition?". The answer to that is "sometimes". The real question is "How do we introduce multiplication to children?"</p>
<p> Professor Devlin doesn't have a good answer for that - and in fact, he weasels out of answering it entirely, which really bugs me. After a long argument about how it's all wrong to teach kids to understand multiplication as repeated addition, and lecturing teachers on how the way that they're teaching is all wrong, he wimps out and says, in essense, "But I don't know anything about teaching, so I can't tell you the right way to do it. All I can do is tell you that you're doing it wrong."</p>
<p> So what are teachers supposed to do? Professor Devlin is very forceful in telling teachers what to do: the last line of his article is: "In the meantime, teachers, please stop telling your pupils that multiplication is repeated addition." But he won't tell those teachers what they should teach their pupils.</p>
<p> You can't tell a teacher to change the way that they're teaching math without<br />
giving them <em>any</em> clue of what the right way to teach it is. What happens in a classroom if the teacher stops using repeated addition to explain multiplication? One of two things will happen. Either the teacher will switch to a different, and<br />
equally incorrect intuition about what multiplication means; or they'll do away with trying to provide any intuition at all.</p>
<p> The right answer is to say that simple multiplication can be understood intuitively in terms of repeated addition. Teachers should do their best to be<br />
clear that it's just an intuition, not the full meaning. Ideally, they should show<br />
multiple ways of understanding it, so that students understand that no one intuition about multiplication is the whole truth. But given a choice between teaching children no intuition, and teaching them a pretty good beginners intuition, I'll take the latter.</p>
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<span><a title="View user profile." href="https://scienceblogs.com/goodmath" lang="" about="https://scienceblogs.com/goodmath" typeof="schema:Person" property="schema:name" datatype="" xml:lang="">goodmath</a></span>
<span>Fri, 07/25/2008 - 09:01</span>
Fri, 25 Jul 2008 13:01:11 +0000goodmath92610 at https://scienceblogs.com